Kinetic theory for electrostatic waves due to transverse velocity shears

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Abstract
A kinetic theory in the form of an integral equation is provided to study the electrostatic oscillations in a collisionless plasma immersed in a uniform magnetic field and a nonuniform transverse electric field. In the low temperature limit (kyρi ≪1, where ky is the wave vector in the y direction and ρi is the ion gyroradius) the dispersion differential equation is recovered for the transverse Kelvin–Helmholtz modes for arbitrary values of k∥, where k∥ is the component of the wave vector in the direction of the external magnetic field assumed in the z direction. For higher temperatures (kyρi>1) the ion‐cyclotron‐like modes described earlier in the literature by Ganguli, Lee, and Palmadesso [Phys. Fluids 2 8, 761 (1985)] are recovered. In this article the integral equation is reduced to a second‐order differential equation and a study is made of the kinetic Kelvin–Helmholtz and the ion‐cyclotron‐like modes that constitute the two branches of oscillation in a magnetized plasma including a transverse inhomogeneous dc electric field.